Question

General solutions of Cos (x-45)= Sin(30-x)

Answer 1

cos(45-x)≡sin(x+45) (All angles are in degrees)

[Proof of this identity: cos(45-x)=cos(45)cos(x)+sin(45)sin(x).

But sin(45)=cos(45), so cos(45-x)=sin(45)cos(x)+cos(45)sin(x)=sin(45+x), or sin(x+45).]

So sin(x+45)=sin(30-x), therefore x+45=30-x+360n where n is an integer.

2x=30-45+360n=360n-15, x=180n-7.5, x=172.5, 352.5, 532.5, etc.

Also, x+45=180-(30-x)+360n=150+x+360n

But the x's cancel out so: 45=150+360n, -105=360n which is false, because n has to be an integer, so x=180n-7.5 are the only solutions.